UPSC NDA I 2021 Mathematics Solved Paper

Concept:The order of a matrix is given by $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.The total number of elements in a matrix of order $m \times n$ is $m \times n$.So, to form different matrices using a fixed set of elements, the product of rows and columns must equal the total number of elements available.Explanation:List all prime numbers less than 30: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.Count them: there are $\mathbf{10}$ elements in total.We need matrices of different orders such that each matrix uses exactly these 10 elements once (the order determines the arrangement).Thus, the product $m \times n$ must be $10$.Find all pairs of positive integers $(m,n)$ whose product is $10$: $(1,10)$, $(2,5)$, $(5,2)$, $(10,1)$.Each pair gives a distinct order: $1 \times 10$, $2 \times 5$, $5 \times 2$, and $10 \times 1$.No other orders are possible because other factor pairs (like $10 \times 1$ already counted) do not exist for integer $m,n \ge 1$.Therefore, exactly 4 different matrix orders can be formed using the given prime numbers.Answer:$\boxed{4}$ (Option A)